Find k from a power-sum condition: If α, β are roots of x^2 − 8x + k = 0 and α^2 + β^2 = 40, find k.

Introduction / Context:
Power sums of roots can be expressed via elementary symmetric sums using identities. Given α + β and αβ from the quadratic, compute α^2 + β^2 and match it to the stated value to solve for k.

Given Data / Assumptions:

• Equation: x^2 − 8x + k = 0.
• α + β = 8, αβ = k.
• α^2 + β^2 = 40.

Concept / Approach:
Use α^2 + β^2 = (α + β)^2 − 2αβ = 8^2 − 2k. Set this equal to 40 and solve for k.

Step-by-Step Solution:

Verification / Alternative check:
With k = 12, the quadratic is x^2 − 8x + 12 = 0 with roots 2 and 6. Then α^2 + β^2 = 4 + 36 = 40, confirming.

Why Other Options Are Wrong:

• 14, 10, 16: Each gives α^2 + β^2 values different from 40 when substituted into 64 − 2k.

Common Pitfalls:
Mixing up the identity and writing 64 + 2k by mistake. Always remember α^2 + β^2 = (α + β)^2 − 2αβ.

Final Answer: